Carathéodory–Jacobi–Lie theorem
# Carathéodory–Jacobi–Lie theorem

The Carathéodory–Jacobi–Lie theorem is a theorem in symplectic geometry which generalizes Darboux's theorem.

## Statement

Let M be a 2n-dimensional symplectic manifold with symplectic form ω. For p ∈ M and r ≤ n, let $f_1,$ $f_2,$ ..., $f_r$ be smooth functions defined on an open neighborhood V of p whose differentials are linearly independent at each point, or equivalently

    $df_{1}(p)\wedge \ldots \wedge df_{r}(p)\neq 0,$

where ${f_i,$ $f_j}$ = 0. (In other words, they are pairwise in involution.) Here {–,–} is the Poisson bracket. Then there are functions $f_{r+1},$ ..., $f_n,$ $g_1,$ $g_2,$ ..., $g_n$ defined on an open neighborhood U ⊂ V of p such that $(f_i,$ $g_i)$ is a symplectic chart of M, i.e., ω is expressed on U as

    $\omega =\sum _{i=1}^{n}df_{i}\wedge dg_{i}.$

## Applications

As a direct application we have the following. Given a Hamiltonian system as $(M,\omega ,H)$ where M is a symplectic manifold with symplectic form $\omega$ and H is the Hamiltonian function, around every point where $dH\neq 0$ there is a symplectic chart such that one of its coordinates is H.

